GOVT. D.B. GIRLS’ P.G. (AUTONOMOUS) COLLEGE RAIPUR CHHATTISGARH FACULTY OF MATHEMATICS SYLLABUS OF M.Sc. MATHEMATICS 2014-15 IIIrd SEMESTER Signature of Chairman D:\533580595.doc Signature of Member (Subject) Page 1 of 16 M.Sc. MATHEMATICS IIIrd SEMESTER Theory Part A Marks No. Title Theory Total Test Seminar Max. 80 Min. 16 Max. 10 Min. 2 Max. 10 Min. 2 100 Paper-I Integration theory and Functional 80 Analysis(I) Paper-II Partial Differential Equations 80 and Mechanics (I) 80 16 10 2 10 2 100 Paper-III Fundamentals of Computer Science 80 70 14 -- 100 -- -- -- 70 Paper-IV Operations Research (I) 80 16 10 2 10 2 100 Paper-V Programming in C (with ANSI features) (I) 70 14 -- -- -- -- 70 80 Practical Part B Name of The Practical Marks Practical-I Object Oriented Programming and Data Structure Practical-II Programming in C NAME Min. 6 30 6 APPROVED BY THE BOARD OF STUDIES ON IN THE CAPACITY OF Dr. MADHU SHRIVASTAVA SIGNATURE CHAIRMAN Dr. PUSHPA KAUSHIK SUBJECT EXPERT Dr. GUPTESHWAR GUPTA SUBJECT EXPERT Mrs. RASHMI SENGUPTA MEMBER OF THE DEPARTMENT Signature of Chairman D:\533580595.doc Max. 30 Signature of Member (Subject) Page 2 of 16 DEPARTMENT OF MATHEMATICS CLASS M.Sc.IIIrd SEMESTER MATHEMATICS SESSION 2014-15 PAPER -I Integration Theory and Functional Analysis (I) Maximum Marks: 80 Minimum Marks: 16 Number of Units : V Unit-I Signed measure. Hahn decomposition theorem, mutually singular measures. Radon-Nikodym theorem. Labesgue decomposition. Riesz representation theorem. Extension theorem (Caratheodory). Unit-II Lebesgue-Stieltjes integral, product measures, Fubini’s theorem. Differentiation and Integration. Decomposition into absolutely continuous and singular parts. Unit-III Baire sets. Baire measure, continuous functions with compact support. Regularity of measures on locally compact spaces. Integration of continuous functions with compact support, Riesz-Markoff theorem. Functional Analysis : Unit-IV Normed linear spaces. Banach spaces and examples. Quotient space of normed linear spaces and its completeness, equivalent norms. Riesz Lemma, basic properties of finite dimensional normed linear spaces and compactness. Unit-V Weak convergence and bounded linear transformations, normed linear spaces of bounded linear transformations, dual spaces with examples. Books Recommended : 1. P.R. Halmos, Measure Theory, Van Nostrand, Princeton, 1950. Signature of Chairman D:\533580595.doc Signature of Member (Subject) Page 3 of 16 2. B.Choudhary and S.Nanda, Functional Analysis with Applications. Wiley Eastern Ltd. 1989. 3. H.L. Royden, Real Analysis, Macmillan Publishing Co. Inc., New York, 4’h Edition, 1993. References 1. S.K. Berberian, Measure and integration, Chelsea Publishing Company, New York, 1965. 2. G. de Barra, Measure Theory and Integration, Wiley Eastern Limited, 1981. 3. P.K. Jain and V.P. Gupta, Lebesgue Measure and Integration, New Age International (P) Limited, New Delhi, 2000. 4. Richard L. Wheeden and Antoni Zygmund, Measure and Integral : An Introduction to Real Analysis, Marcel Dekker Inc. 1977. 5. J.H. Williamson, Lebesgue Integration, Holt Rinehart and Winston, Inc. New York. 1962. 6. T.G. Hawkins, Lebesgue’s Theory of Integration: Its Origins and Development, Chelsea, New York, 1979. 7. K.R. Parthasarathy, Introduction to Probability and Measure, Macmillan Company of India Ltd., Delhi, 1977. 8. R.G. Bartle, The Elements of Integration, John Wiley & Sons, Inc. New York, 1966. 9. Serge Lang, Analysis I & II, Addison-Wesley Publishing Company, Inc. 1967. 10. Inder K. Rana, An Introduction to Measure and Integration, Narosa Publishing House, Delhi, 1997. 11. Walter Rudin, Real & Complex Analysis, Tata McGraw-Hill Publishing. 12. Edwin Hewitt and Korl Stromberg, Real and Abstract Analysis, SpringerVerlag, New York. 13. Edwin Hewitt and Kenneth A. Ross, Abstract Harmonic Analysis, Vol. 1, Springer-Verlag, 1993. 14. G. Bachman and L. Narici, Functional Analysis, Academic Press, 1966. 15. N. Dunford and J.T. Schwartz, Linear Operators, Part I, Interscience, New York, 1958. 16. R.E. Edwards, Functional Analysis, Holt Rinehart and Winston, New York, 1965. 17. C. Goffman and G. Pedrick, First Course in Functional Analysis, Prentice Hall of India, New Delhi, 1987. 18. P.K. Jain, O.P. Ahuja and Khalil Ahmad, Functional Analysis, New Age International (P) Ltd. & Wiley Eastern Ltd., New Delhi, 1997. Signature of Chairman D:\533580595.doc Signature of Member (Subject) Page 4 of 16 19. R.B. Holmes, Geometric Functional Analysis and its Applications, SpringerVerlag, 1975. 20. K.K. Jha, Functional Analysis, Students’ Friends, 1986. 21. L.V. Kantorovich and G.P. Akilov, Functional Analysis, Pergamon Press, 1982. 22. E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, New York, 1978. 23. B.K. Lahiri, Elements of Functional Analysis, The World Press Pvt. Ltd., Calcutta, 1994. 24. A.H.Siddiqui, Functional Analysis with Applications, Tata McGraw-Hill Publishing Company Ltd. New Delhi 25. B.V. Limaye, Functional Analysis, Wiley Eastern Ltd. 26. L.A. Lustenik and V.J. Sobolev, Elements of Functional Analysis, Hindustan Publishing Corporation, New Delhi, 1971. 27. G.F. Simmons, Introduction to Topology and Modern Analysis, McGraw-Hill Book Company, New York, 1963. 28. A.E. Taylor, Introduction to Functional Analysis, John Wiley and Sons, New York, 1958. 29. K.Yosida, Functional Analysis, 3'” edition Springer-Verlag, New York, 1971. 30. J.B. Conway, A Course in Functional Analysis, Springer-Verlag, New York, 1990. 31. Walter Rudin, Functional Analysis, Tata McGraw-Hill Publishing Company Ltd., New Delhi, 1973. 32. A. Wilansky, Functional Analysis, Blaisdell Publishing Co., 1964. 33. J. Tinsley Oden & Leszek F. Dernkowicz, Applied Functional Analysis, CRC Press Inc., 1996. Signature of Chairman D:\533580595.doc Signature of Member (Subject) Page 5 of 16 DEPARTMENT OF MATHEMATICS CLASS M.Sc.IIIrd SEMESTER MATHEMATICS SESSION 2014-15 PAPER -II Partial Differential Equations and Mechanics (I) Maximum Marks: 80 Minimum Marks: 16 Partial Differential Equations: Number of Units : V Unit-I Examples of PDE. Classification.Transport Equation-Initial value Problem. Nonhomogeneous Equation. Laplace’s Equation-Fundamental Solution, Mean Value Formulas, Properties of Harmonic Functions, Green’s Function, Energy Methods. Unit-II Heat Equation-Fundamental Solution, Mean Value Formula, Properties of Solutions, Energy Methods. Wave Equation-Solution by Spherical Means, Non-homogeneous Equations, Energy Methods. Analytical Dynamics: Unit-III Generalized coordinates. Holonomic and Non-holonomic systems. Scleronomic and Rheonomic sytems. Generalized potential. Lagrange’s equations of first kind. Lagrange’s equations of second kind. Uniqueness of solution. Energy equation for conservative fields. Hamilton’s variables. Donkin’s theorem. Hamilton canonical equations. Cyclic coordinates. Routh’s equations. Unit-IV Poisson’s Bracket. Poisson’s Identity. Jacobi-Poisson Theorem. Motivating problems of calculus of variations, Shortest distance. Minimum surface of revolution. Brachistochrone problem. Isoperimetric problem. Geodesic. Fundamental lemma of calculus of variations. Euler’s equation for one dependent function and its generalization to (1) ‘n’ dependent functions, (ii) higher order derivatives. Conditional extremum under geometric constraints and under integral constraints. Gravitation: Unit-V Attraction and potential of rod, disc, spherical shells and sphere. Surface integral of normal attraction (application & Gauss’ theorem). Laplace and Poisson equations. Work done by selfattracting systems. Distributions for a given potential. Equipotential surfaces. Surface and solid harmonics. Surface density in terms of surface harmonics. Books Recommended : Signature of Chairman D:\533580595.doc Signature of Member (Subject) Page 6 of 16 1. L.C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, Volume 19, AMS, 1998. 2. F. Gantmacher, Lectures in Analytic Mechanics, MIR Publishers, Moscow, 1975. 3. R.C.Mondal, Classical Mechanics, Prentice Hall of India 4. S.L. Loney, An Elementary Treatise on Statics, Kalyani Publishers, New Delhi, 1979. References 1. Books on Partial differential equation by 1.N. Sneddon, F. John, P. Prasad and R. Ravindran, Amarnath etc. 2. A.S. Ramsey, Dynamics Part II, The English Language Book Society and Cambridge University Press, 1972. 3. H. Goldstein, Classical Mechanics (2nd edition), Narosa Publishing House, New 4. 5. 6. 7. Delhi. I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice Hall. Narayan Chandra Rana & Pramod Sharad Chandra Joag, Classical Mechanics, Tata McGraw Hill, 1991. Louis N. Hand and Janet D. Finch, Analytical Mechanics, Cambridge University Press, 1998. A.S. Ramsey, Newtonian Gravitation, The English Language Book Society and the Cambridge University Press. DEPARTMENT OF MATHEMATICS CLASS M.Sc.IIIrd SEMESTER Signature of Chairman D:\533580595.doc Signature of Member (Subject) Page 7 of 16 MATHEMATICS SESSION 2014-15 PAPER-III Fundamentals of Computer Science-Theory and Practical (Object Oriented Programming and Data Structure) Maximum Marks: 70 Number of Units : V Minimum Marks: 14 Unit-I Object Oriented Programming-Classes and Scope, nested classes, pointer class members; Class initialization, assignment and destruction. Unit-II Overloaded functions and operators; Templates including class templates; class inheritance and virtual functions. Unit-III Data Structures-Analysis of algorithms, q, W, 0, o, w notations ; Sequential and linked representations, Lists, Stacks, and queues; Unit-IV Trees: Binary tree- search tree implementation, B-tree (concept only); Unit-V Sorting: Insertion sort, shell sort, quick-sort, heap sort and their analysis; Hashing-open and closed. Books Recommended : 1. S.B. Lipman, J. Lajoi: C++ Primer, Addison Wesley. 2. B. Stroustrup; The C++ Programming Language, Addison Wesley. 3. C.J. Date : Introduction to Database Systems, Addison Wesley. 4. C. Ritchie: Operating Systems-Incorporating UNIX and Windows, BPB Publications. 5. M.A. Weiss, Data Structures and Algorithm Analysis in C++, Addison Wesley. Signature of Chairman D:\533580595.doc Signature of Member (Subject) Page 8 of 16 DEPARTMENT OF MATHEMATICS CLASS M.Sc.IIIrd SEMESTER MATHEMATICS SESSION 2014-15 PAPER -IV Operations Research (I) Maximum Marks: 80 Minimum Marks: 16 Number of Units : V Unit-I Operations Research and its Scope. Necessity of Operations Research in Industry. Linear Programming-Simplex Method. Theory of the Simplex Method. Duality and Sensitivity Analysis. Unit-II Other Algorithms for Linear Programmrng-Dual Simplex Method. Unit-III Parametric Linear Programming. Upper Bound Technique. Interior Point Algorithm. Linear Goal Programming. Unit-IV Transportation and Assignment Problems. Unit-V Network Analysis-Shortest Path Problem. Minimum Spanning Tree Probiem. Maximum Flow I Problem. Minimum Cost Flow Problem. Network Simplex Method. Project Planning and Control I with PERT-CPM. Signature of Chairman D:\533580595.doc Signature of Member (Subject) Page 9 of 16 Books Recommended : 1. F.S. Hillier and G.J. Ueberman. Introduction to Operations ResBareft (Sixth Edition), McGraw Hill International Edition, Industrial Engineering Series, 1995. (This book comes with a CD containing tutorial software). 2. G. Hadley, Linear Programming, Narosa Publishing House, 1995. 3. G. Hadly, Nonlinear and Dynamic Programming, Addison-Wesley, Reading Mass. 4. H.A. Taha, Operations Research -An introduction, Macmillan Publishing Co., Inc., New Yark. 5. Kanti Swarup, P.K. Gupta and Man Mohan, Operations Research, Sultan Chand & Sons, New Delhi 6. Mokhtar S. Bazaraa, John J. Jarvis and Hanif D. Sherali, Linear Programming and Network flows, John Wiley I ‘ & Sons, New York, 1990. 1. Ltd., New Delhi. 2. Prem Kumar Gupla and D.S. Hira, Operations Research-An Introduction. S. Cliand & Company Ltd., New Delhi. 3. N.S. Kambo, Mathematical Programming Techniques, Affiliated East-West Press Pvt. Ltd., New Delhi, Madras 4. R.K. Rathy, An Introduction to Fluid Dynamics, Oxford and IBH Publishing Company, New Delhi, 1976. 5. A.D. Young, Boundary Layers, AIAA Education Series, Washington DC, 1989. 6. S.W. Yuan, Foundations of Fluid Mechanics, Prentice Hall of India Private Limited, New Delhi, 1976. 7. UNDOSystems Products (Visit websHe htlp://www.Hndo.com/productsf.html) (i) UNDO (the linear programming solver) (ii) UNDO Callable Library (the premier optimisation engine) (iii) LINGO (the linear, non-linear, and integer programming solver with mathematical modelling language) (i) What’s Best I (the spreadssheet add-in that solves linear, non-linear, and integer problems). All the above four products are bundled into one package to form the Solver Suite. For more details about any of the four products one has to click on its name. (i) Optimisation Modelling with UNDO (8" edition) by Linus Schrage. (ii) Optimisation Modelling with LINGO by Unus Schrage. More details available on the Related Boo~s page York, 1979. Signature of Chairman D:\533580595.doc Signature of Member (Subject) Page 10 of 16 DEPARTMENT OF MATHEMATICS CLASS M.Sc.IIIrd SEMESTER MATHEMATICS SESSION 2014-15 PAPER -V Programming in C (with ANSI features) Theory and Practical Maximum Marks: 70 Minimum Marks: 14 Number of Units : V Unit-I An overview of programming. Programming language, Classification. C Essentials-Program Development. Functions. Anatomy of a C Function.Variables and Constants. Expressions. Assignment Statements. Formatting Source Files. Continuation Character. The Preprocessor. Unit-II Scalar Data Types-Declarations, Different Types of Integers. Different kinds of Integer Constants. Floating-Point Types. Initialization. Mixing Types. Explicit Conversions-Casts. Enumeration Types. The Void Data Type. Typedefs. Finding the Address of an object. Pointers. Unit-III Control Flow-Conditional Branching. The Switch Statement. Looping. Nested Loops. The break and continue Statements. The goto statement. Infinite Loops. Unit-IV Operators and Expressions-Precedence and Associativity. Unary Plus and Minus operators. Binary Arithmetic Operators. Arithmetic Assignment Operators. Increment and Decrement Operators. Comma Operator. Relational Operators. Logical Operators. Bit - Manipulation Operators. Bitwise Assignment Operators. Cast Operator. Size of Operators. Conditional Operator. Memory Operators. Unit-V Arrays -Declaring an Array. Arrays and Memory. Initializing Arrays. Encryption and Decryption. Books Recommended : Signature of Chairman D:\533580595.doc Signature of Member (Subject) Page 11 of 16 1. 2. 3. Peter A. Darnell and Philip E. Margolis, C: A Software Engineering Approach, Narosa Publishing House (Springer International Student Edition) 1993. Samuel P. Harkison and Gly L. Steele Jr., C : A Reference Manual, 2nd Edition, Prentice Hall, 1984. Brian W. Kernighan & Dennis M. Ritohie, The C Programme Language, 2nd Edition (ANSI Features), Prentice Hall 1989. DEPARTMENT OF MATHEMATICS Signature of Chairman D:\533580595.doc Signature of Member (Subject) Page 12 of 16 CLASS M.Sc.IIIrd SEMESTER SESSION 2014-15 PRACTICAL-I Object Oriented Programming and Data Structure Maximum Marks: 30 Minimum Marks: 06 Distribution of Marks Sessional - 05 Practical - 20 Viva - 05 List of Practicals: C++ program 1. Write a Program to reverse the digit of number and compare them and also find the sum of digits of number. 2. Write a Program using inline function to find multiplication and division of any two numbers. 3. Write a Program to create class BOOK having data members book_no , book_title , price and member functions to read the details of book ,total cost of books and function purchase to ask the user to input the number of copies to be purchased. 4. Write a Program to create class COMPLEX having member function to store and show real and imaginary part of number and also show addition of two complex numbers. 5. Write a Program to calculate area of circle, rectangle, triangle using function overloading concept. 6. Write a Program to change the sign and increase the value by one of given number using operator overloading concept. 7. Write a Program for implementation of copy constructor. 8. Create a class circle with data member radius; provide member function to calculate area. Derive a class sphere from class circle; provide member function to calculate volume. 9. Write a Program to create class student with data member roll_no; provide member function to get and put roll_no. Derive a class test provide member functions to read and show the marks of two subjects. Again Derive a class result from class test to display the result of student including the total marks obtained by student. 10. Write a Program that illustrate use of virtual function. 11. Write a Program to find biggest number among three numbers using pointer and function. 12. Write a Program using function Template to swap different type of numbers. Data Structure Programs 1. Write a Program to insert an element and replace an element by new element at desired Signature of Chairman Signature of Member (Subject) D:\533580595.doc Page 13 of 16 position in an array. 2. Write a Program to search the required element in an array. 3. Write a Program to find the largest element from an array. 4. Write a Program to perform Bubble Sorting in an array. 5. Write a Program to perform Insertion Sorting in an array. 6. Write a Program to perform Quick Sort in an array. 7. Write a Program to push and pop element on Stack. 8. Write a Program to insert and show element from Queue. 9. Write a program to perform Heap Sort in an array. 10. Write a Program to search the required element in Binary Tree DEPARTMENT OF MATHEMATICS CLASS M.Sc.IIIrd SEMESTER Signature of Chairman D:\533580595.doc Signature of Member (Subject) Page 14 of 16 SESSION 2014-15 PRACTICAL-II Programming in C Maximum Marks: 30 Minimum Marks: 06 Distribution of Marks Sessional - 05 Practical - 20 Viva - 05 List of Practicals: INPUT/OUTPUT & OPERATORS /EXPRESSIONS 1. 2. 3. 4. WAP to apply all arithmetic symbols. WAP to find simple interest. WAP to calculate the sum and average of five numbers. Write a program to swap the two numbers a) using third variable b) without using third variable 5. WAP to convert Fahrenheit into centigrade and vice versa. 6. Write a Program to calculate area and perimeter of a circle 7. WAP compute surface ,area and volume of a cube. 8. Write a program to enter any two numbers and display greater number using ternary operator. BRANCHING 9. Write a program to check the given number is even or odd. a) using ternary operator. b) using only if statements. c) using if else statements. d) using goto statements. 10. Write a program to enter any three numbers and display greater number. a) using ternary operator. b) using if statements. c) using if else statements. 11. Write a program to check the given year is leap or not. 12. WAP to calculate the power of an integer. 13. WAP to calculate sum of natural numbers. 14. Write a program to check the number is perfect or not. 15. Write a program to check the square is perfect or not. 16. Write a program to calculate electricity bill using switch case. 17. Write a program of roots of a Quadratic Equation. LOOPING 18. Print the following a) 2,4,6,8,…………….n. ( the even series) b) 1,3,5,7………………n. (the odd series) c) 2,1,4,3,6,………..n,n-1. d) Write a program to display Fibonacci series. 19. WAP to display a table of any number. 20. Write a program to check the number is Armstrong or not using for loop. 21. Write a program to calculate the sum of all digits of given number. 22. Write a program to find factorial of any number Signature of Chairman Signature of Member (Subject) D:\533580595.doc Page 15 of 16 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. a) using for loop b) using while loop c) using do-while loop Write a Program to find out the prime numbers between 1 to 100. Write a program to generate the following pattern: a) * * * b.) * c) 1 * * * * 12 * * * * 1 2 3 ARRAYS WAP to print 1D array elements. WAP to sum of a 1D array elements. Write a program to find out the greatest and smallest element in 1-D array. WAP to print 2D array elements. WAP to find the sum all 2D array (matrix) elements. Write a program to create the addition of two matrixes. Write a program to display a diagonal matrix. Write a program to multiply the two matrices. Write a program to find out the greatest and smallest element in 2-D array. Signature of Chairman D:\533580595.doc Signature of Member (Subject) Page 16 of 16

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